Continuity

When Godzilla enters the Japanese harbor he uses his atomic breath to blow up all of the resistance along the shore. When he starts using it he turns to the left, facing the camera, but the breath still goes to the right on the shore.

Godzilla's appearance keeps changing due to different special effect techniques being used. In particular, the giant animatronic model (known as "Cybot") mostly used for closeups has a drastically different look from the suit: the neck and arms are much thinner, the snout rounder, the ears more prominent, and the eyes and general expression also look different. It also has more robotic movements, as it's being controlled by motors instead of a suit actor.

Crew or equipment visible

As Dr. Hayashida's sonic oscillator starts luring Godzilla out of Tokyo to the volcano, there's a POV shot from Goro and Naoko's perspective that clearly shows the wire controlling Godzilla's tail. This almost appears to be an unintentional homage to the scene in the original Gojira, where Godzilla's tail appears outside the window of a partially demolished building (the one in which Raymond Burr's character was reporting from in the American version), and the wire is similarly visible.

Plot holes

In the American version, the American general and his peers keep
talking about giving support to the Japanese, but the only real visible thing they "help" with is the stopping of the Russian nuclear missile.

Revealing mistakes

In the composite shots of the gathered crowd looking at the temporarily downed Godzilla, there is a blurry line where the two parts of the shot meet, so the people in the front of the crowd apparently disappear as they go closer to Godzilla. When they start running away, some of the people seemingly pop into existence.